Boundary Value Problem with Regular SIngurarlty and Helgason–Okamoto Conjecture

First we review on the definition of boundary values of solutions of linear differential equations. The most fundamental example is the concept of hyperfunctions, which are the boundary values of holomorphic functions. The space of hyperfunctions on R is defined by J3 (JR) = 0(C+)©0(CL)/0(O. Here C± = {z = x + iyt=C; ±3>>0} and 0(U) is the space of holomorphic functions on a complex manifold U. Therefore any hyperfunction f(x) on B equals the sum of the boundary values of solutions of the equation dzu(z) =1/2(d/dx + id/dy)u(z) = 0: f(x) =u+(x + £0) *_ (x -zO) (u± (*) e 0 (C±) ).